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Mirrors > Home > MPE Home > Th. List > mapssfset | Structured version Visualization version GIF version |
Description: The value of the set exponentiation (𝐵 ↑m 𝐴) is a subset of the class of functions from 𝐴 to 𝐵. (Contributed by AV, 10-Aug-2024.) |
Ref | Expression |
---|---|
mapssfset | ⊢ (𝐵 ↑m 𝐴) ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapfset 8791 | . . 3 ⊢ (𝐵 ∈ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑m 𝐴)) | |
2 | eqimss2 4002 | . . 3 ⊢ ({𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑m 𝐴) → (𝐵 ↑m 𝐴) ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐵 ∈ V → (𝐵 ↑m 𝐴) ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}) |
4 | reldmmap 8777 | . . . 4 ⊢ Rel dom ↑m | |
5 | 4 | ovprc1 7397 | . . 3 ⊢ (¬ 𝐵 ∈ V → (𝐵 ↑m 𝐴) = ∅) |
6 | 0ss 4357 | . . 3 ⊢ ∅ ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵} | |
7 | 5, 6 | eqsstrdi 3999 | . 2 ⊢ (¬ 𝐵 ∈ V → (𝐵 ↑m 𝐴) ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}) |
8 | 3, 7 | pm2.61i 182 | 1 ⊢ (𝐵 ↑m 𝐴) ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 {cab 2710 Vcvv 3444 ⊆ wss 3911 ∅c0 4283 ⟶wf 6493 (class class class)co 7358 ↑m cmap 8768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8770 |
This theorem is referenced by: (None) |
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